Numerical Methods For Stochastic Differential Equations

Stochastic differential equations(SDEs)arise in a variety of fields in sciences such as physics,chemistry,biology and engineering.Unfortunately,for many stochastic differential equations,the analytical solution is not acquirable and the numerical solution becomes a dispensable approach.In this thesis,we investigate effective numerical integration for two typical classes of stochastic ordinary differential equations.For systems of stochastic dif-ferential equations of Stratonovich type we have set up stochastic partitioned Runge-Kutta(SPRK)methods as well as a rooted tree theory,and for the stochastic oscillator we have developed a new family of zero-dissipative methods.This thesis is divided into four chapters:Chapter 1 gives the research background of this thesis and introduces the basic elements of stochastic differential equations.In Chapter 2,we study the Runge-Kutta methods for stochastic ordinary differential equations of Stratonovich type.The rooted tree theory,order conditions for SRK methods are presented.Apart from several explicit methods in the literature,a three-stage explicit SRK method and two two-stage implicit SRK methods are constructed.The results of numerical experiments show that the average errors produced by the SRK methods constructed are much smaller than that of the Euler-Maruyama(EM)method.The error of the four-stage SRK4s method of strong order 1.5 of Burrage is the smallest among the explicit methods selected.Moreover,the errors produced by the two two-stage implicit methods of strong order 1,the imSRK2sl and imSRK2s2 methods,are very close to that of SRK4s.Comparing with the exact orbit,the numerical orbit produced by imSRK2s2 is the most consistent while the EM method exhibits the opposite.In Chapter 3,inspired by the SRK methods in Chapter 2,we consider the SPRK methods for systems of stochastic ordinary differential equations of Stratonovich type with additive noise.A new stochastic four-colored rooted tree theory is developed by which order conditions for strong order 1 and 1.5 are derived,and three practical methods are obtained.The results of numerical experiments show that the mean errors of the SPRK and SRK methods applied are smaller than the EM method regardless the noise intensity.And when noise grows large enough,the SPRK method is superior to the SRK method.For the Stratonovich L-V system,the SPRK and SRK methods can preserve the energy of the system in the long time simulation.In Chapter 4,we investigate the numerical integrators for the linear stochastic oscillators with additive noise.We have proved that the partitioned Euler-Maruyama(PEM)method and the implicit midpoint(IM)method can preserve the linear growth property of the second moment of the exact solution,while the Euler-Maruyama(EM)method and the backward Euler-Maruyama(BEM)method can not.Based on this,we propose the notion of stochastic dissipation.We have shown that the EM method,BEM method and the PEM method all have dissipation order 1,and the IM method has dissipation order 2.We have constructed new adapted methods,such as the adapted Euler-Maruyama(AEM)method,the adapted backward Euler-Maruyama(ABEM)method and the adapted mid-point(AMP)method,and we have shown that all the three adapted methods not only have the same linear growth property as the exact solution,but also are zero dissipative.Finally we summarize the main contributions of this work and outlook some further topics for future work.